Integral dari $$$e^{\frac{x^{2}}{8}}$$$

Temukan $$$\int e^{\frac{x^{2}}{8}}\, dx$$$.

Misalkan $$$u=\frac{\sqrt{2} x}{4}$$$.

Kemudian $$$du=\left(\frac{\sqrt{2} x}{4}\right)^{\prime }dx = \frac{\sqrt{2}}{4} dx$$$ (langkah-langkah dapat dilihat di »), dan kita memperoleh $$$dx = 2 \sqrt{2} du$$$.

Dengan demikian,

$${\color{red}{\int{e^{\frac{x^{2}}{8}} d x}}} = {\color{red}{\int{2 \sqrt{2} e^{u^{2}} d u}}}$$

Terapkan aturan pengali konstanta $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ dengan $$$c=2 \sqrt{2}$$$ dan $$$f{\left(u \right)} = e^{u^{2}}$$$:

$${\color{red}{\int{2 \sqrt{2} e^{u^{2}} d u}}} = {\color{red}{\left(2 \sqrt{2} \int{e^{u^{2}} d u}\right)}}$$

Integral ini (Fungsi Galat Imajiner) tidak memiliki bentuk tertutup:

$$2 \sqrt{2} {\color{red}{\int{e^{u^{2}} d u}}} = 2 \sqrt{2} {\color{red}{\left(\frac{\sqrt{\pi} \operatorname{erfi}{\left(u \right)}}{2}\right)}}$$

Ingat bahwa $$$u=\frac{\sqrt{2} x}{4}$$$:

$$\sqrt{2} \sqrt{\pi} \operatorname{erfi}{\left({\color{red}{u}} \right)} = \sqrt{2} \sqrt{\pi} \operatorname{erfi}{\left({\color{red}{\left(\frac{\sqrt{2} x}{4}\right)}} \right)}$$

Oleh karena itu,

$$\int{e^{\frac{x^{2}}{8}} d x} = \sqrt{2} \sqrt{\pi} \operatorname{erfi}{\left(\frac{\sqrt{2} x}{4} \right)}$$

Tambahkan konstanta integrasi:

$$\int{e^{\frac{x^{2}}{8}} d x} = \sqrt{2} \sqrt{\pi} \operatorname{erfi}{\left(\frac{\sqrt{2} x}{4} \right)}+C$$

$$$\int e^{\frac{x^{2}}{8}}\, dx = \sqrt{2} \sqrt{\pi} \operatorname{erfi}{\left(\frac{\sqrt{2} x}{4} \right)} + C$$$A